Calculus

Problem 11501

Position function: $x=2 \pi t+\bar{c} \cos 2 \pi t, 0 \leq t \leq 5$ .
a) When is the particle moving left/right? b) When is it at rest? c) Find average rate of change in 5s. d) Estimate velocity each second.

See Solution

Problem 11502

A particle's position is $x=2 \pi t+\cos 2 \pi t, 0 \leq t \leq 5$ . When does it move left/right and when is it at rest?

See Solution

Problem 11503

Find critical points of $f(x)=3 x^{3}-14 x^{2}+7 x$ on $[0,5]$ and extreme values $f_{\min}, f_{\max}$ to 3 decimals.

See Solution

Problem 11504

Find $d y / d x$ using implicit differentiation for the equation $2 x y - y^{2} = 1$ .

See Solution

Problem 11505

Find the critical numbers for the function $f(x)=3 x^{4}+8 x^{3}-210 x^{2}+102$ .

See Solution

Problem 11506

Find the production level $x$ that minimizes the marginal cost function $C(x) = x^2 - 140x + 9000$ and determine the minimum cost.

See Solution

Problem 11507

Find the derivative of $s=t^{4} \tan t-\sqrt{t}$ .

See Solution

Problem 11508

Find $\frac{d y}{d x}$ for $y=\tan u$ and $u=-13 x+20$ .

See Solution

Problem 11509

Find $\frac{d y}{d x}$ for $y=\tan u$ where $u=-13 x+20$ . Use the chain rule.

See Solution

Problem 11510

Un cube de glace de $27 \mathrm{~cm}^{3}$ fond selon $\frac{d V}{d t}=-0.6 \mathrm{x}^{2}$ . Trouvez $\frac{d x}{d t}$ et le volume après 7 min.

See Solution

Problem 11511

Find the velocity and speed of a stone dropped from a 216-ft cliff on Mars, where $s(t) = -6t^2 + 216$ .

See Solution

Problem 11512

A particle's position is $x=2 \pi t+\overline{c o s} 2 \pi t$ for $0 \leq t \leq 5$ . Find when it moves left/right, is at rest, average rate of change, and velocity per second.

See Solution

Problem 11513

Is the integral $\int_{-\infty}^{\infty} \frac{x}{x^{2}+9} d x$ convergent or divergent? If convergent, find its value.

See Solution

Problem 11514

Determine if the integral $\int_{0}^{\infty} \frac{d x}{(x-5)^{\frac{1}{3}}}$ is convergent or divergent. If convergent, find its value.

See Solution

Problem 11515

Find the second derivative of $y=\frac{x^{4}+1}{x^{2}}$ .

See Solution

Problem 11516

Find the derivative of $y$ using logarithmic differentiation, where $y=\frac{x \sqrt{x^{3}+2}}{(x+4)^{1/3}}$ .

See Solution

Problem 11517

Find the tangent line equation for $y=\frac{x^{2}}{2}$ at $x=5$ .

See Solution

Problem 11518

Find the derivative of the function $y=\frac{x^{2}-3 x+2}{x^{7}-2}$ .

See Solution

Problem 11519

Find the interval(s) where the profit function, derived from $C(x)=0.17 x^{2}-0.00008 x^{3}$ and $R(x)=0.746 x^{2}-0.0002 x^{3}$ , is increasing.
A. The profit function is increasing on the interval(s) $\square$ . B. The profit function is never increasing.

See Solution

Problem 11520

Find the limit as $x$ approaches 4 for $3+\sqrt{x h(x)}$ given $\lim _{x \rightarrow 4} g(x)=-9$ and $\lim _{x \rightarrow 4} h(x)=3$ . Round to three decimal places.

See Solution

Problem 11521

Find $y_{0}$ such that points $P=(1,0)$ and $Q=(2,y_{0})$ lie on the graph where the slope is $8x-5$ .

See Solution

Problem 11522

Write $y=f(u)$ and $u=g(x)$ for $y=(-3x+10)^{5}$ , then find $\frac{dy}{dx}$ as a function of $x$ .

See Solution

Problem 11523

Find the intervals where the drug concentration given by $K(t)=\frac{16 t}{t^{2}+16}$ is increasing and decreasing.

See Solution

Problem 11524

Find the intervals where the function $A(x)=0.004 x^{3}-0.05 x^{2}+0.18 x+0.05$ is increasing or decreasing on $[0,5]$ .

See Solution

Problem 11525

Verify that $\frac{d y}{d x}=\frac{1}{19 y^{18}}$ for the curve $x=y^{19}$ and find $\frac{d^{2} y}{d x^{2}}$ .

See Solution

Problem 11526

Calculate the force variation $d F$ for a passenger of mass $m=63.0 \mathrm{~kg}$ on a ride with radius $r=5.30 \mathrm{~m}$ and speed $v=2.00 \mathrm{~m/s}$ due to changes in (a) radius $d r$ , (b) speed $d v$ , and (c) period $d T$ . Provide derivatives and units.

See Solution

Problem 11527

Check if the integral $\int_{4}^{\infty} \frac{d x}{\ln x - 1}$ converges.

See Solution

Problem 11528

Find the critical numbers, increasing intervals, and decreasing intervals for $f(x)=\frac{4}{3} x^{3}-14 x^{2}-72 x-24$ . First, compute $f^{\prime}(x)$ .

See Solution

Problem 11529

For the function $f(x)=6 x^{4}+16 x^{3}-288 x^{2}+33$ , find critical numbers and intervals of increase/decrease.

See Solution

Problem 11530

Find the time $t$ in $0 \leq t \leq 6$ when a particle with acceleration $a(t)=2t-10$ and initial velocity 21 ft/sec changes direction.

See Solution

Problem 11531

Find all values of $c$ in the interval $\left(\frac{1}{4}, 24\right)$ where $f^{\prime}(c)=0$ for $f(x)=x+6 x^{-1}$ .

See Solution

Problem 11532

Find all values of $c$ in $(2,10)$ for the function $f(x)=\frac{x^{2}}{6 x-10}$ where $f^{\prime}(c)=0$ .

See Solution

Problem 11533

Detective Daniels investigates a murder using Newton's Law of Cooling. Find the cooling rate $k$ , the function $T(t)$ , and time of death.
a. Calculate $k$ using body temperatures and room temp. b. Write $T(t)$ with initial temp $T_{0} = 98.6^{\circ} \mathrm{F}$ . c. Solve $T(t) = 77.9^{\circ} \mathrm{F}$ for $t$ to find time of death.

See Solution

Problem 11534

Evaluate the integral: $\int_{-1}^{7} x \sqrt[3]{x+1} \, dx$

See Solution

Problem 11535

Find the function $f(x)$ given that $f^{\prime \prime}(x)=2x+6\sin(x)$ , $f(0)=2$ , and $f^{\prime}(0)=4$ .

See Solution

Problem 11536

Find the unique anti-derivative $F$ of $f(x)=(4-x)^{2}-\frac{2}{(4-x)^{2}}$ with $F(0)=0$ , and calculate $F(3)$ .

See Solution

Problem 11537

Find $y_{0}$ so that points $P=(1,3)$ and $Q=(2, y_{0})$ lie on the graph where the slope is $8x - 7$ .

See Solution

Problem 11538

For an even function $f(x)$ with $\int_{0}^{2} f(x) d x=\frac{8}{3}$ , find the integrals:
15. $\int_{-2}^{0} f(x) d x$ ,
16. $\int_{-2}^{2} f(x) d x$ ,
17. $\int_{-2}^{0} 3 f(x) d x$ .
For an odd function $f(x)$ with $\int_{0}^{2} f(x) d x=\frac{8}{3}$ , find the integrals:
18. $\int_{-2}^{0} f(x) d x$ ,
19. $\int_{-2}^{2} f(x) d x$ ,
20. $\int_{-2}^{0} 3 f(x) d x$ .

See Solution

Problem 11539

関数 $y=-x^{2}$ で、 $x$ が 1 から 4 まで増加する時の変化の割合を求めなさい。

See Solution

Problem 11540

Evaluate the integral $\int_{R} e^{-x-y} d x d y$ for the region $R$ in the first quadrant where $x+y \leq 1$ .

See Solution

Problem 11541

Find the function $f(x)$ given that $f^{\prime \prime}(x)=8 x+10 \sin (x)$ , $f(0)=3$ , and $f^{\prime}(0)=3$ . $f(x)=$

See Solution

Problem 11542

Find the position $s(t)$ of the particle given $v(t)=7 \sin(t)-6 \cos(t)$ and $s(0)=5$ .

See Solution

Problem 11543

Find $f^{\prime}(x)$ given $f^{\prime \prime}(x)=6 x-3$ , $f^{\prime}(-1)=1$ , and $f(-1)=-5$ . Also, find $f(2)$ .

See Solution

Problem 11544

Find where $H(r)=\frac{300}{1+0.06 r^{2}}$ is increasing/decreasing and calculate $H'(r)$ .

See Solution

Problem 11545

Find the position of a particle at time $t=13$ given $a(t)=12t+18$ , $s(0)=2$ , and $v(0)=17$ .

See Solution

Problem 11546

Find the antiderivative $F(t)$ of $f(t)=4 \sec ^{2}(t)-6 t^{3}$ with $F(0)=0$ . Calculate $F(5)$ .

See Solution

Problem 11547

Find values of $c$ in $(2,4)$ where $f^{\prime}(c)=0$ for $f(x)=\frac{x^{2}}{3 x-4}$ . $c=$

See Solution

Problem 11548

Find $a$ so that the line $x=a$ bisects the area under $y=\frac{1}{x^{2}}$ from $1$ to $4$ . Also, find $b$ for $y=b$ .

See Solution

Problem 11549

Find where the function $H(r) = \frac{300}{1 + 0.06 r^2}$ is increasing and decreasing with respect to the mortgage rate $r$ .

See Solution

Problem 11550

Find all values of $c$ in the interval $\left(\frac{1}{4}, 28\right)$ where $f^{\prime}(c)=0$ for $f(x)=x+7 x^{-1}$ .
$c=$

See Solution

Problem 11551

Find $f(5)$ for the function with $f^{\prime \prime}(x)=7x+10\sin(x)$ , $f(0)=4$ , $f^{\prime}(0)=3$ .

See Solution

Problem 11552

Revolve the area between $y=\sin(x)$ and $y=0$ from $0$ to $\pi$ around the $x$ -axis. Find the volume and properties.

See Solution

Problem 11553

Find the maximum concentration $C(t)=\frac{0.014 t}{t^{2}+3 t+3}$ .
Give $C_{\max }$ to five decimal places and $t$ (time) to three decimal places.

See Solution

Problem 11554

A cone-shaped cup is 16 cm deep with a 4 cm radius. Water flows in at 2 cm³/sec. Find rates of height and radius change at given depths.

See Solution

Problem 11555

Find $f^{\prime}(x)$ given $f^{\prime \prime}(x)=3x-3$ , $f^{\prime}(-2)=6$ , and $f(-2)=2$ . Also, find $f(2)$ .

See Solution

Problem 11556

Find the minimum height of the catenary $y=11 \cosh \left(\frac{x}{11}\right)-5.4$ for $x \in [-3,3]$ . Report to one decimal place.

See Solution

Problem 11557

A snowball's volume increases at $10 \mathrm{in}^{3}/\mathrm{min}$ . Find the radius and surface area increase rates at $36 \pi \mathrm{in}^{3}$ .

See Solution

Problem 11558

Find the limit: $\lim _{x \rightarrow 0}(\sin (x+9)-\sin (9))=$

See Solution

Problem 11559

How long does it take for a bacteria population to double with a growth rate of $2.4\%$ per hour? Round to the nearest hundredth.

See Solution

Problem 11560

An observer at point A sees balloon B rising from C at 3 m/sec. Find rates of change when y=50 for x, area, and θ.

See Solution

Problem 11561

Find $f^{\prime}(x)$ given $f^{\prime \prime}(x)=5x+3$ , $f^{\prime}(-2)=3$ , and $f(-2)=4$ . Also, find $f(4)$ .

See Solution

Problem 11562

Find $f^{\prime}(x)$ given $f^{\prime \prime}(x)=4 x+4$ , $f^{\prime}(-1)=4$ , and $f(-1)=-5$ . Also find $f(2)$ .

See Solution

Problem 11563

Find $f^{\prime}(x)$ given $f^{\prime \prime}(x)=4 x+4$ , $f^{\prime}(-1)=4$ , and $f(-1)=-5$ . Then find $f(2)$ .

See Solution

Problem 11564

Find the derivative of $\sin\left(\frac{\pi}{2} - x\right)$ .

See Solution

Problem 11565

An observer at point A watches balloon B rise from point C at 3 m/s, 100 m away. Find rates of change when $y=50$ .

See Solution

Problem 11566

Find $f^{\prime}(x)$ from $f^{\prime \prime}(x)=9x-2$ , given $f^{\prime}(-1)=-2$ , $f(-1)=-3$ . Then, find $f(3)$ .

See Solution

Problem 11567

Find the horizontal asymptote of $f(x)=\frac{4-3x-2x^{2}}{3x^{2}+x-1}$ and the $x$ value where it touches.

See Solution

Problem 11568

Find the limit as $x$ approaches $0$ of $\frac{\sin (x+9)-\sin (9)}{x}$ .

See Solution

Problem 11569

Find $f^{\prime}(x)$ from $f^{\prime \prime}(x)=7x-5$ , given $f^{\prime}(0)=-4$ and $f(0)=6$ . Also, find $f(4)$ .

See Solution

Problem 11570

Find $f^{\prime}(x)$ from $f^{\prime \prime}(x)=4 x-6$ with $f^{\prime}(-3)=4$ , $f(-3)=-6$ . Then, find $f(3)$ .

See Solution

Problem 11571

Find $f^{\prime}(x)$ from $f^{\prime \prime}(x)=9x-1$ with $f^{\prime}(-3)=3$ , $f(-3)=-4$ , then find $f(3)$ .

See Solution

Problem 11572

Given $f(2)=5$ , $f^{\prime}(2)=-2$ , and $f^{\prime}(3)=1$ , which statement is true about local extrema?

See Solution

Problem 11573

What method finds the dose $a$ that maximizes blood pressure $b$ ? Choose one: a, b, c, or d.

See Solution

Problem 11574

Find the local maximum of $f(x)=x^{3}-9 x^{2}-48 x+52$ . Choose from: $x=-2$ , $x=e$ , $x=5$ , $x=8$ .

See Solution

Problem 11575

Calculate the integral: $\int \frac{1+\cos (4 t)}{2} d t$

See Solution

Problem 11576

Find the limit: $\lim _{x \rightarrow-\infty}\left(\sqrt{4 x^{2}+3 x}+2 x\right)$ .

See Solution

Problem 11577

Find $f^{\prime}(x)$ given $f^{\prime \prime}(x)=2 x+3$ , $f^{\prime}(-3)=4$ , and $f(-3)=-5$ . Also, find $f(2)$ .

See Solution

Problem 11578

Gewinnfunktion: $G(x)=-0,5 x^{3}+2 x^{2}+2,5 x-6$ für $x \in[0 ; 5]$ .
3.1. Berechne $G(0)$ , $G(1)$ , $G(4,5)$ und interpretiere. 3.2. Finde die Produktionsmenge für maximalen Gewinn und den Gewinn. 3.3. Bestimme das Intervall [a; b] für Gewinn.

See Solution

Problem 11579

Berechne die Werte $G(0)$ , $G(1)$ und $G(4.5)$ für die Gewinnfunktion $G(x)=-0,5x^3+2x^2+2,5x-6$ und finde die Produktionsmenge mit maximalem Gewinn. Bestimme auch die Gewinnintervalle.

See Solution

Problem 11580

Find the zeros, extrema, and inflection points of $f(x)=2 x^{2} e^{-0.1 x}$ .

See Solution

Problem 11581

Find $\frac{d y}{d x}$ for the function $y=2 x+\frac{1}{\sqrt{4 x}}$ .

See Solution

Problem 11582

Find the marginal revenue from renting the 20th apartment given $R(x)=6x^3-8x^2-\sqrt{x}$ . Choose the correct option.

See Solution

Problem 11583

Berechne den Extrempunkt der Funktion $f(x)=2 x^{2} \cdot e^{-0,1 x}$ .

See Solution

Problem 11584

Éric lance une balle de $2 \mathrm{~kg}$ d'un édifice de $356 \mathrm{~m}$ à $24 \mathrm{~m/s}$ . Trouvez l'énergie, vitesses et hauteur à différentes étapes.

See Solution

Problem 11585

Find the derivative of the function $f(x)=\frac{x}{(x+1)^{2}}$ and verify that $f^{\prime}(x)=\frac{1-x}{(x+1)^{3}}$ .

See Solution

Problem 11586

Ein Unternehmen produziert Farben. a) Bestimme die Erlösfunktion $E$ und zeige, dass $E$ bei 33 Einheiten maximiert wird. b) Zeige, dass die Kostenfunktion $K$ keine Extremstellen hat. c) Benenne die Graphen der Funktionen $p, E, K, G$ . d) Zeige, dass $G(x)=-2 x^{3}+85 x^{2}+300 x-3375$ gilt und berechne den maximalen Gewinn. e) Prüfe, ob 30 Einheiten weiterhin den Gewinn maximieren, wenn die Fixkosten auf 4000 steigen.

See Solution

Problem 11587

Given the function $f(x)=\frac{x}{(x+1)^{2}}$ , find $f^{\prime}(x)$ , $f^{\prime \prime}(x)$ , and sketch $f(x)$ using curve sketching.

See Solution

Problem 11588

Find the derivative of $y=\sqrt{4 x^{3}-7}$ .

See Solution

Problem 11589

Find the derivative of $y=x \sqrt{x+1}$ . What is $\frac{d y}{d x}$ ?

See Solution

Problem 11590

Bestimme die Intervalle, in denen die Funktionen $f(x)=x^{2}-x-6$ , $f(x)=x^{3}+x$ , und $f(x)=\frac{1}{9} x^{3}-x$ monoton sind.

See Solution

Problem 11591

Find the differential $d y$ for $y=\sqrt{3+x^{2}}$ and evaluate it when $d y=1$ , $d x=-0.2$ .

See Solution

Problem 11592

Find the slope of the tangent line to $f(x)=x^{2}-3$ at $(-2,1)$ using $\lim _{h \rightarrow 0}\left(\frac{f(a+h)-f(a)}{h}\right)$ and write its equation.

See Solution

Problem 11593

Find the instantaneous rate of change of $f(x)=\sqrt{x}$ at $x=9$ using $\lim _{x \rightarrow 9}\left(\frac{f(x)-f(9)}{x-9}\right)$ .

See Solution

Problem 11594

Find the derivative $f^{\prime}(4)$ for the function $f(x)=-\sqrt{x}-\frac{\sqrt{x^{3}}}{6}$ .

See Solution

Problem 11595

Find $f^{\prime}(-3)$ for the function $f(x)=-2 x e^{x}$ . Provide the answer in simplest form.

See Solution

Problem 11596

Find $f^{\prime}(-3)$ for the function $f(x)=-2 x e^{x}$ . Simplify your answer with no negative exponents.

See Solution

Problem 11597

Find $f^{\prime}(1)$ for the function $f(x)=\frac{e^{x}}{3 x^{2}+2}$ . Simplify your answer without negative exponents.

See Solution

Problem 11598

Find these limits: a. $\lim _{x \rightarrow 5}\left(\frac{x^{2}-3 x-10}{x-5}\right)$ , b. $\lim _{x \rightarrow \infty}\left(\frac{5-x}{x^{4}-x^{3}}\right)$ , c. $\lim _{x \rightarrow-\infty}\left(7 x^{2}+3 x^{3}+18000 x+6\right)$ .

See Solution

Problem 11599

Find the vertical asymptote(s) of the graph of $y=\frac{\ln (1-x)}{x+1}$ .

See Solution

Problem 11600

Bestimmen Sie die Ableitung von $f(x)=\frac{2}{\sqrt{x}}+\frac{\sqrt{x}}{2}$ .

See Solution

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 244 245 246 247 248 249 250 251 252 253 254 255 256 257 258 259 260 261 262 263 264 265 266 267 268 269 270 271 272 273 274 275 276 277 278 279 280 281 282 283 284 285 286 287 288 289 290 291 292 293 294 295 296 297 298 299 300 301 302 303 304 305 306 307 308 309 310 311 312 313 314 315 316 317 318 319 320 321 322 323 324 325 326 327 328 329 330 331 332 333 334 335 336 337

Calculus

Problem 11501

Position function: x=2πt+cˉcos⁡2πt,0≤t≤5x=2 \pi t+\bar{c} \cos 2 \pi t, 0 \leq t \leq 5x=2πt+cˉcos2πt,0≤t≤5. a) When is the particle moving left/right? b) When is it at rest? c) Find average rate of change in 5s. d) Estimate velocity each second.

Problem 11502

A particle's position is x=2πt+cos⁡2πt,0≤t≤5x=2 \pi t+\cos 2 \pi t, 0 \leq t \leq 5x=2πt+cos2πt,0≤t≤5. When does it move left/right and when is it at rest?

Problem 11503

Find critical points of f(x)=3x3−14x2+7xf(x)=3 x^{3}-14 x^{2}+7 xf(x)=3x3−14x2+7x on [0,5][0,5][0,5] and extreme values fmin⁡,fmax⁡f_{\min}, f_{\max}fmin​,fmax​ to 3 decimals.

Problem 11504

Find dy/dxd y / d xdy/dx using implicit differentiation for the equation 2xy−y2=12 x y - y^{2} = 12xy−y2=1.

Problem 11505

Find the critical numbers for the function f(x)=3x4+8x3−210x2+102f(x)=3 x^{4}+8 x^{3}-210 x^{2}+102f(x)=3x4+8x3−210x2+102.

Problem 11506

Find the production level xxx that minimizes the marginal cost function C(x)=x2−140x+9000C(x) = x^2 - 140x + 9000C(x)=x2−140x+9000 and determine the minimum cost.

Problem 11507

Find the derivative of s=t4tan⁡t−ts=t^{4} \tan t-\sqrt{t}s=t4tant−t​.

Problem 11508

Find dydx\frac{d y}{d x}dxdy​ for y=tan⁡uy=\tan uy=tanu and u=−13x+20u=-13 x+20u=−13x+20.

Problem 11509

Find dydx\frac{d y}{d x}dxdy​ for y=tan⁡uy=\tan uy=tanu where u=−13x+20u=-13 x+20u=−13x+20. Use the chain rule.

Problem 11510

Un cube de glace de 27 cm327 \mathrm{~cm}^{3}27 cm3 fond selon dVdt=−0.6x2\frac{d V}{d t}=-0.6 \mathrm{x}^{2}dtdV​=−0.6x2. Trouvez dxdt\frac{d x}{d t}dtdx​ et le volume après 7 min.

Problem 11511

Find the velocity and speed of a stone dropped from a 216-ft cliff on Mars, where s(t)=−6t2+216s(t) = -6t^2 + 216s(t)=−6t2+216.

Problem 11512

A particle's position is x=2πt+cos‾2πtx=2 \pi t+\overline{c o s} 2 \pi tx=2πt+cos2πt for 0≤t≤50 \leq t \leq 50≤t≤5. Find when it moves left/right, is at rest, average rate of change, and velocity per second.

Problem 11513

Is the integral ∫−∞∞xx2+9dx\int_{-\infty}^{\infty} \frac{x}{x^{2}+9} d x∫−∞∞​x2+9x​dx convergent or divergent? If convergent, find its value.

Problem 11514

Determine if the integral ∫0∞dx(x−5)13\int_{0}^{\infty} \frac{d x}{(x-5)^{\frac{1}{3}}}∫0∞​(x−5)31​dx​ is convergent or divergent. If convergent, find its value.

Problem 11515

Find the second derivative of y=x4+1x2y=\frac{x^{4}+1}{x^{2}}y=x2x4+1​.

Problem 11516

Find the derivative of yyy using logarithmic differentiation, where y=xx3+2(x+4)1/3y=\frac{x \sqrt{x^{3}+2}}{(x+4)^{1/3}}y=(x+4)1/3xx3+2​​.

Problem 11517

Find the tangent line equation for y=x22y=\frac{x^{2}}{2}y=2x2​ at x=5x=5x=5.

Problem 11518

Find the derivative of the function y=x2−3x+2x7−2y=\frac{x^{2}-3 x+2}{x^{7}-2}y=x7−2x2−3x+2​.

Problem 11519

Problem 11520

Find the limit as xxx approaches 4 for 3+xh(x)3+\sqrt{x h(x)}3+xh(x)​ given lim⁡x→4g(x)=−9\lim _{x \rightarrow 4} g(x)=-9limx→4​g(x)=−9 and lim⁡x→4h(x)=3\lim _{x \rightarrow 4} h(x)=3limx→4​h(x)=3. Round to three decimal places.

Problem 11521

Find y0y_{0}y0​ such that points P=(1,0)P=(1,0)P=(1,0) and Q=(2,y0)Q=(2,y_{0})Q=(2,y0​) lie on the graph where the slope is 8x−58x-58x−5.

Problem 11522

Write y=f(u)y=f(u)y=f(u) and u=g(x)u=g(x)u=g(x) for y=(−3x+10)5y=(-3x+10)^{5}y=(−3x+10)5, then find dydx\frac{dy}{dx}dxdy​ as a function of xxx.

Problem 11523

Find the intervals where the drug concentration given by K(t)=16tt2+16K(t)=\frac{16 t}{t^{2}+16}K(t)=t2+1616t​ is increasing and decreasing.

Problem 11524

Find the intervals where the function A(x)=0.004x3−0.05x2+0.18x+0.05A(x)=0.004 x^{3}-0.05 x^{2}+0.18 x+0.05A(x)=0.004x3−0.05x2+0.18x+0.05 is increasing or decreasing on [0,5][0,5][0,5].

Problem 11525

Verify that dydx=119y18\frac{d y}{d x}=\frac{1}{19 y^{18}}dxdy​=19y181​ for the curve x=y19x=y^{19}x=y19 and find d2ydx2\frac{d^{2} y}{d x^{2}}dx2d2y​.

Problem 11526

Problem 11527

Check if the integral ∫4∞dxln⁡x−1\int_{4}^{\infty} \frac{d x}{\ln x - 1}∫4∞​lnx−1dx​ converges.

Problem 11528

Find the critical numbers, increasing intervals, and decreasing intervals for f(x)=43x3−14x2−72x−24f(x)=\frac{4}{3} x^{3}-14 x^{2}-72 x-24f(x)=34​x3−14x2−72x−24. First, compute f′(x)f^{\prime}(x)f′(x).

Problem 11529

For the function f(x)=6x4+16x3−288x2+33f(x)=6 x^{4}+16 x^{3}-288 x^{2}+33f(x)=6x4+16x3−288x2+33, find critical numbers and intervals of increase/decrease.

Problem 11530

Find the time ttt in 0≤t≤60 \leq t \leq 60≤t≤6 when a particle with acceleration a(t)=2t−10a(t)=2t-10a(t)=2t−10 and initial velocity 21 ft/sec changes direction.

Problem 11531

Find all values of ccc in the interval (14,24)\left(\frac{1}{4}, 24\right)(41​,24) where f′(c)=0f^{\prime}(c)=0f′(c)=0 for f(x)=x+6x−1f(x)=x+6 x^{-1}f(x)=x+6x−1.

Problem 11532

Find all values of ccc in (2,10)(2,10)(2,10) for the function f(x)=x26x−10f(x)=\frac{x^{2}}{6 x-10}f(x)=6x−10x2​ where f′(c)=0f^{\prime}(c)=0f′(c)=0.

Problem 11533

Problem 11534

Evaluate the integral: ∫−17xx+13 dx\int_{-1}^{7} x \sqrt[3]{x+1} \, dx∫−17​x3x+1​dx

Problem 11535

Find the function f(x)f(x)f(x) given that f′′(x)=2x+6sin⁡(x)f^{\prime \prime}(x)=2x+6\sin(x)f′′(x)=2x+6sin(x), f(0)=2f(0)=2f(0)=2, and f′(0)=4f^{\prime}(0)=4f′(0)=4.

Problem 11536

Find the unique anti-derivative FFF of f(x)=(4−x)2−2(4−x)2f(x)=(4-x)^{2}-\frac{2}{(4-x)^{2}}f(x)=(4−x)2−(4−x)22​ with F(0)=0F(0)=0F(0)=0, and calculate F(3)F(3)F(3).

Problem 11537

Find y0y_{0}y0​ so that points P=(1,3)P=(1,3)P=(1,3) and Q=(2,y0)Q=(2, y_{0})Q=(2,y0​) lie on the graph where the slope is 8x−78x - 78x−7.

Problem 11538

Problem 11539

関数 y=−x2y=-x^{2}y=−x2 で、xxx が 1 から 4 まで増加する時の変化の割合を求めなさい。

Problem 11540

Evaluate the integral ∫Re−x−ydxdy\int_{R} e^{-x-y} d x d y∫R​e−x−ydxdy for the region RRR in the first quadrant where x+y≤1x+y \leq 1x+y≤1.

Problem 11541

Find the function f(x)f(x)f(x) given that f′′(x)=8x+10sin⁡(x)f^{\prime \prime}(x)=8 x+10 \sin (x)f′′(x)=8x+10sin(x), f(0)=3f(0)=3f(0)=3, and f′(0)=3f^{\prime}(0)=3f′(0)=3. f(x)= f(x)= f(x)=

Problem 11542

Position function: $x=2 \pi t+\bar{c} \cos 2 \pi t, 0 \leq t \leq 5$ .
a) When is the particle moving left/right? b) When is it at rest? c) Find average rate of change in 5s. d) Estimate velocity each second.

A particle's position is $x=2 \pi t+\cos 2 \pi t, 0 \leq t \leq 5$ . When does it move left/right and when is it at rest?

Find critical points of $f(x)=3 x^{3}-14 x^{2}+7 x$ on $[0,5]$ and extreme values $f_{\min}, f_{\max}$ to 3 decimals.

Find $d y / d x$ using implicit differentiation for the equation $2 x y - y^{2} = 1$ .

Find the critical numbers for the function $f(x)=3 x^{4}+8 x^{3}-210 x^{2}+102$ .

Find the production level $x$ that minimizes the marginal cost function $C(x) = x^2 - 140x + 9000$ and determine the minimum cost.

Find the derivative of $s=t^{4} \tan t-\sqrt{t}$ .

Find $\frac{d y}{d x}$ for $y=\tan u$ and $u=-13 x+20$ .

Find $\frac{d y}{d x}$ for $y=\tan u$ where $u=-13 x+20$ . Use the chain rule.

Un cube de glace de $27 \mathrm{~cm}^{3}$ fond selon $\frac{d V}{d t}=-0.6 \mathrm{x}^{2}$ . Trouvez $\frac{d x}{d t}$ et le volume après 7 min.

Find the velocity and speed of a stone dropped from a 216-ft cliff on Mars, where $s(t) = -6t^2 + 216$ .

A particle's position is $x=2 \pi t+\overline{c o s} 2 \pi t$ for $0 \leq t \leq 5$ . Find when it moves left/right, is at rest, average rate of change, and velocity per second.

Is the integral $\int_{-\infty}^{\infty} \frac{x}{x^{2}+9} d x$ convergent or divergent? If convergent, find its value.

Determine if the integral $\int_{0}^{\infty} \frac{d x}{(x-5)^{\frac{1}{3}}}$ is convergent or divergent. If convergent, find its value.

Find the second derivative of $y=\frac{x^{4}+1}{x^{2}}$ .

Find the derivative of $y$ using logarithmic differentiation, where $y=\frac{x \sqrt{x^{3}+2}}{(x+4)^{1/3}}$ .

Find the tangent line equation for $y=\frac{x^{2}}{2}$ at $x=5$ .

Find the derivative of the function $y=\frac{x^{2}-3 x+2}{x^{7}-2}$ .

Find the limit as $x$ approaches 4 for $3+\sqrt{x h(x)}$ given $\lim _{x \rightarrow 4} g(x)=-9$ and $\lim _{x \rightarrow 4} h(x)=3$ . Round to three decimal places.

Find $y_{0}$ such that points $P=(1,0)$ and $Q=(2,y_{0})$ lie on the graph where the slope is $8x-5$ .

Write $y=f(u)$ and $u=g(x)$ for $y=(-3x+10)^{5}$ , then find $\frac{dy}{dx}$ as a function of $x$ .

Find the intervals where the drug concentration given by $K(t)=\frac{16 t}{t^{2}+16}$ is increasing and decreasing.

Find the intervals where the function $A(x)=0.004 x^{3}-0.05 x^{2}+0.18 x+0.05$ is increasing or decreasing on $[0,5]$ .

Verify that $\frac{d y}{d x}=\frac{1}{19 y^{18}}$ for the curve $x=y^{19}$ and find $\frac{d^{2} y}{d x^{2}}$ .

Check if the integral $\int_{4}^{\infty} \frac{d x}{\ln x - 1}$ converges.

Find the critical numbers, increasing intervals, and decreasing intervals for $f(x)=\frac{4}{3} x^{3}-14 x^{2}-72 x-24$ . First, compute $f^{\prime}(x)$ .

For the function $f(x)=6 x^{4}+16 x^{3}-288 x^{2}+33$ , find critical numbers and intervals of increase/decrease.

Find the time $t$ in $0 \leq t \leq 6$ when a particle with acceleration $a(t)=2t-10$ and initial velocity 21 ft/sec changes direction.

Find all values of $c$ in the interval $\left(\frac{1}{4}, 24\right)$ where $f^{\prime}(c)=0$ for $f(x)=x+6 x^{-1}$ .

Find all values of $c$ in $(2,10)$ for the function $f(x)=\frac{x^{2}}{6 x-10}$ where $f^{\prime}(c)=0$ .

Evaluate the integral: $\int_{-1}^{7} x \sqrt[3]{x+1} \, dx$

Find the function $f(x)$ given that $f^{\prime \prime}(x)=2x+6\sin(x)$ , $f(0)=2$ , and $f^{\prime}(0)=4$ .

Find the unique anti-derivative $F$ of $f(x)=(4-x)^{2}-\frac{2}{(4-x)^{2}}$ with $F(0)=0$ , and calculate $F(3)$ .

Find $y_{0}$ so that points $P=(1,3)$ and $Q=(2, y_{0})$ lie on the graph where the slope is $8x - 7$ .

関数 $y=-x^{2}$ で、 $x$ が 1 から 4 まで増加する時の変化の割合を求めなさい。

Evaluate the integral $\int_{R} e^{-x-y} d x d y$ for the region $R$ in the first quadrant where $x+y \leq 1$ .

Find the function $f(x)$ given that $f^{\prime \prime}(x)=8 x+10 \sin (x)$ , $f(0)=3$ , and $f^{\prime}(0)=3$ . $f(x)=$

Find the position $s(t)$ of the particle given $v(t)=7 \sin(t)-6 \cos(t)$ and $s(0)=5$ .

Find $f^{\prime}(x)$ given $f^{\prime \prime}(x)=6 x-3$ , $f^{\prime}(-1)=1$ , and $f(-1)=-5$ . Also, find $f(2)$ .

Find where $H(r)=\frac{300}{1+0.06 r^{2}}$ is increasing/decreasing and calculate $H'(r)$ .

Find the position of a particle at time $t=13$ given $a(t)=12t+18$ , $s(0)=2$ , and $v(0)=17$ .

Find the antiderivative $F(t)$ of $f(t)=4 \sec ^{2}(t)-6 t^{3}$ with $F(0)=0$ . Calculate $F(5)$ .

Find values of $c$ in $(2,4)$ where $f^{\prime}(c)=0$ for $f(x)=\frac{x^{2}}{3 x-4}$ . $c=$

Find $a$ so that the line $x=a$ bisects the area under $y=\frac{1}{x^{2}}$ from $1$ to $4$ . Also, find $b$ for $y=b$ .

Find where the function $H(r) = \frac{300}{1 + 0.06 r^2}$ is increasing and decreasing with respect to the mortgage rate $r$ .

Find all values of $c$ in the interval $\left(\frac{1}{4}, 28\right)$ where $f^{\prime}(c)=0$ for $f(x)=x+7 x^{-1}$ .
$c=$

Find $f(5)$ for the function with $f^{\prime \prime}(x)=7x+10\sin(x)$ , $f(0)=4$ , $f^{\prime}(0)=3$ .

Revolve the area between $y=\sin(x)$ and $y=0$ from $0$ to $\pi$ around the $x$ -axis. Find the volume and properties.

Find the maximum concentration $C(t)=\frac{0.014 t}{t^{2}+3 t+3}$ .
Give $C_{\max }$ to five decimal places and $t$ (time) to three decimal places.

Find $f^{\prime}(x)$ given $f^{\prime \prime}(x)=3x-3$ , $f^{\prime}(-2)=6$ , and $f(-2)=2$ . Also, find $f(2)$ .

Find the minimum height of the catenary $y=11 \cosh \left(\frac{x}{11}\right)-5.4$ for $x \in [-3,3]$ . Report to one decimal place.

A snowball's volume increases at $10 \mathrm{in}^{3}/\mathrm{min}$ . Find the radius and surface area increase rates at $36 \pi \mathrm{in}^{3}$ .

Find the limit: $\lim _{x \rightarrow 0}(\sin (x+9)-\sin (9))=$

How long does it take for a bacteria population to double with a growth rate of $2.4\%$ per hour? Round to the nearest hundredth.

Find $f^{\prime}(x)$ given $f^{\prime \prime}(x)=5x+3$ , $f^{\prime}(-2)=3$ , and $f(-2)=4$ . Also, find $f(4)$ .

Find $f^{\prime}(x)$ given $f^{\prime \prime}(x)=4 x+4$ , $f^{\prime}(-1)=4$ , and $f(-1)=-5$ . Also find $f(2)$ .

Find $f^{\prime}(x)$ given $f^{\prime \prime}(x)=4 x+4$ , $f^{\prime}(-1)=4$ , and $f(-1)=-5$ . Then find $f(2)$ .

Find the derivative of $\sin\left(\frac{\pi}{2} - x\right)$ .

An observer at point A watches balloon B rise from point C at 3 m/s, 100 m away. Find rates of change when $y=50$ .

Find $f^{\prime}(x)$ from $f^{\prime \prime}(x)=9x-2$ , given $f^{\prime}(-1)=-2$ , $f(-1)=-3$ . Then, find $f(3)$ .

Find the horizontal asymptote of $f(x)=\frac{4-3x-2x^{2}}{3x^{2}+x-1}$ and the $x$ value where it touches.

Find the limit as $x$ approaches $0$ of $\frac{\sin (x+9)-\sin (9)}{x}$ .

Find $f^{\prime}(x)$ from $f^{\prime \prime}(x)=7x-5$ , given $f^{\prime}(0)=-4$ and $f(0)=6$ . Also, find $f(4)$ .

Find $f^{\prime}(x)$ from $f^{\prime \prime}(x)=4 x-6$ with $f^{\prime}(-3)=4$ , $f(-3)=-6$ . Then, find $f(3)$ .

Find $f^{\prime}(x)$ from $f^{\prime \prime}(x)=9x-1$ with $f^{\prime}(-3)=3$ , $f(-3)=-4$ , then find $f(3)$ .

Given $f(2)=5$ , $f^{\prime}(2)=-2$ , and $f^{\prime}(3)=1$ , which statement is true about local extrema?

What method finds the dose $a$ that maximizes blood pressure $b$ ? Choose one: a, b, c, or d.

Find the local maximum of $f(x)=x^{3}-9 x^{2}-48 x+52$ . Choose from: $x=-2$ , $x=e$ , $x=5$ , $x=8$ .